BHET Meeting
McGill University
2023-12-21
Estimate the total effect of the intervention.
Estimate the contribution of changes in the chemical composition of PM_{2.5} to the overall effect on health outcomes.
Examine alternative pathways and mechanisms that may contribute to the intervention’s impact.
To understand the pathways, mechanisms, and intermediates through which a treatment affects an outcome.
Step 1: Estimate the total effect of T.
Basic idea: understand pathways of effects
Step 2: Estimate how much of the total effect is due to PM_{2.5} vs. other pathways?
X = pre-treatment covariates
T = exposure
M = mediator
W = confounders
Y = outcome
X = pre-treatment covariates
T = exposure
M = mediator
W = confounders
Y = outcome
Total effect: \color{red}{E[Y|T,X]=\beta_{0}+\beta_{1}T +\beta_{2}X}
This equation estimates the total effect of the ban: \color{red}{TE=\beta_{1}(T^{*}-T)} where T^{*} is exposure to ban and T is no exposure.
Estimate two regressions:1 {E[M|T,X]=\beta_{0}+\beta_{1}T +\beta_{2}X} \color{red}E[Y|T,X,M] = \theta_{0} + \theta_{1}T + \theta_{2}M + \theta_{3}TM + \theta_{4}X + \theta_{5}W
Second equation estimates the “Controlled Direct Effect”: \color{red}{CDE=\theta_{1}+\theta_{3}TM}
Assumptions for valid CDE:
Valid NDE and NIE also require:
This effect is the contrast between the counterfactual outcome if the individual were exposed at T=t and the counterfactual outcome if the same individual were exposed at T=t*, with the mediator set to a fixed level M=m.
“By how much would blood pressure change if the policy were implemented and we held PM_{2.5} fixed at m ?“
X = cohort, time FEs
T = policy
M = hours of sleep
W = {empty}
Y = Poor respiratory symptoms
‘Poor respiratory symptoms’ = 1 if frequency of any coughing, wheezing, etc. were “most” or “several” days a week.
| Unique (#) | Missing (%) | Mean | SD | Min | Median | Max | ||
|---|---|---|---|---|---|---|---|---|
| v_id | 50 | 0 | 25.3 | 14.2 | 1.0 | 25.0 | 50.0 | |
| year | 3 | 0 | 2019.4 | 1.2 | 2018.0 | 2019.0 | 2021.0 | |
| cohort_year | 4 | 0 | 2018.6 | 0.9 | 2018.0 | 2018.0 | 2021.0 | |
| treat | 2 | 0 | 0.2 | 0.4 | 0.0 | 0.0 | 1.0 | |
| resp | 2 | 0 | 0.5 | 0.5 | 0.0 | 1.0 | 1.0 | |
| hsleep | 30 | 0 | 7.7 | 2.0 | 1.0 | 8.0 | 20.0 |
logit(Y_{it}) = \alpha^{village}_{v[i]} + \sum_{r=q}^{T} \beta_{r} d_{r} + \sum_{s=r}^{T} \gamma_{s} fs_{t}+ \sum_{r=q}^{T} \sum_{s=r}^{T} \tau_{rt} (d_{r} \times fs_{t})
Simple Average |
|||
|---|---|---|---|
| Est. | 2.5 % | 97.5 % | |
| treat | −0.106 | −0.203 | −0.003 |
Cohort Averages |
||||
|---|---|---|---|---|
| Est. | (S.E. ) | 2.5 % | 97.5 % | |
| ATT(g2019) | −0.158 | (0.067) | −0.284 | −0.023 |
| ATT(g2020) | 0.013 | (0.075) | −0.137 | 0.158 |
| ATT(g2021) | −0.017 | (0.111) | −0.241 | 0.194 |
logit(Y_{it}) = \alpha^{village}_{v[i]} + \sum_{r=q}^{T} \beta_{r} d_{r} + \sum_{s=r}^{T} \gamma_{s} fs_{t}+ \sum_{r=q}^{T} \sum_{s=r}^{T} \tau_{rt} (d_{r} \times fs_{t}) \\ + \delta M_{it} + \sum_{r=q}^{T} \sum_{s=r}^{T} \eta_{rt} (d_{r} \times fs_{t} \times M_{it}) where now we have added:
Total Effect |
CDE |
|||||
|---|---|---|---|---|---|---|
| Est. | 2.5 % | 97.5 % | Est. | 2.5 % | 97.5 % | |
| Untreated | 0.606 | 0.514 | 0.690 | 0.599 | 0.505 | 0.688 |
| Treated | 0.498 | 0.444 | 0.553 | 0.500 | 0.447 | 0.554 |
| Difference | −0.106 | −0.203 | −0.003 | −0.098 | −0.202 | 0.008 |
Minimal evidence of mediation.
Proportion explained: PE = \frac{TE - CDE}{TE} = 0.08
More complicated
Sequential mediators?
Interactions between mediators?
Mediation analysis aims are part of HEI project.
Likely to focus mostly on CDEs.
Tutorials, packages and macros in R, SAS, Stata available.1
Implementation with staggered DiD more likely to require manual implementation.